James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

580 Chapter 8 Further Applications of Integration
7. A spinner from a board game randomly indicates a real number
between 0 and 10. The spinner is fair in the sense that it indicates
a number in a given interval with the same probability as
it indicates a number in any other interval of the same length.
(a) Explain why the function
f sxd −H 0.1
0
if 0 < x < 10
if x , 0 or x . 10
is a probability density function for the spinner’s values.
(b) What does your intuition tell you about the value of the
mean? Check your guess by evaluating an integral.
8. (a) Explain why the function whose graph is shown is a probability
density function.
(b) Use the graph to find the following probabilities:
(i) PsX , 3d (ii) Ps3 < X < 8d
(c) Calculate the mean.
y
0.2
0.1
0 2
y=ƒ
4 6 8 10 x
9. Show that the median waiting time for a phone call to the company
described in Example 4 is about 3.5 minutes.
10. (a) A type of light bulb is labeled as having an average lifetime
of 1000 hours. It’s reasonable to model the probability of
failure of these bulbs by an exponential density function
with mean − 1000. Use this model to find the probability
that a bulb
(i) fails within the first 200 hours,
(ii) burns for more than 800 hours.
(b) What is the median lifetime of these light bulbs?
11. An online retailer has determined that the average time for
credit card transactions to be electronically approved is
1.6 seconds.
(a) Use an exponential density function to find the probability
that a customer waits less than a second for credit card
approval.
(b) Find the probability that a customer waits more than
3 seconds.
(c) What is the minimum approval time for the slowest 5% of
transactions?
12. The time between infection and the display of symptoms
for streptococcal sore throat is a random variable whose
probabililty density function can be approximated by
f std − 1
15,676 t 2 e 20.05t if 0 < t < 150 and f std − 0 otherwise
(t measured in hours).
(a) What is the probability that an infected patient will display
symptoms within the first 48 hours?
(b) What is the probability that an infected patient will not
display symptoms until after 36 hours?
Source: Adapted from P. Sartwell, “The Distribution of Incubation Periods of
Infectious Disease,” American Journal of Epidemiology 141 (1995): 386–94.
13. REM sleep is the phase of sleep when most active dreaming
occurs. In a study, the amount of REM sleep during the first
four hours of sleep was described by a random variable T with
probability density function
1
1600 t if 0 < t < 40
1
f std − µ 20 2 1
1600 t if 40 , t < 80
0 otherwise
where t is measured in minutes.
(a) What is the probability that the amount of REM sleep is
between 30 and 60 minutes?
(b) Find the mean amount of REM sleep.
14. According to the National Health Survey, the heights of adult
males in the United States are normally distributed with mean
69.0 inches and standard deviation 2.8 inches.
(a) What is the probability that an adult male chosen at random
is between 65 inches and 73 inches tall?
(b) What percentage of the adult male population is more than
6 feet tall?
15. The “Garbage Project” at the University of Arizona reports
that the amount of paper discarded by households per week is
normally distributed with mean 9.4 lb and standard deviation
4.2 lb. What percentage of households throw out at least 10 lb
of paper a week?
16. Boxes are labeled as containing 500 g of cereal. The machine
filling the boxes produces weights that are nor mally distributed
with standard deviation 12 g.
(a) If the target weight is 500 g, what is the probability that the
machine produces a box with less than 480 g of cereal?
(b) Suppose a law states that no more than 5% of a manufacturer’s
cereal boxes can contain less than the stated weight
of 500 g. At what target weight should the manufacturer set
its filling machine?
17. The speeds of vehicles on a highway with speed limit
100 kmyh are normally distributed with mean 112 kmyh and
standard deviation 8 kmyh.
(a) What is the probability that a randomly chosen vehicle is
traveling at a legal speed?
(b) If police are instructed to ticket motorists driving
125 kmyh or more, what percentage of motorists are
targeted?
18. Show that the probability density function for a normally distributed
random variable has inflection points at x − 6 .
19. For any normal distribution, find the probability that the random
variable lies within two standard deviations of the mean.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.